Duals of higher real $K$-theories at $p=2$

We study $\mathrm{K}(h)$-local Spanier-Whitehead duality for $C_{2^n}$-equivariant Lubin-Tate spectra, $E_h$, at the prime $2$ and heights $h$ divisible by $2^{n-1}$. We determine a $C_{2^n}$-equivariant equivalence $D{E}_h\simeq {\mathrm{\Sigma }}^{-V_h}{E}_h$, for an explicit $C_{2^n}$-representation, $V_h$. We then study the $\mathrm{R}\mathrm{O}(C_{2^n})$-periodicities of $E_h$ at some low heights. With these ingredients, we determine the self-duality of some higher real $K$-theories up to a specified suspension shift, at some low-heights. In particular, we show that $D{E}_4^{h{C}_8}\simeq {\mathrm{\Sigma }}^{112}{E}_4^{h{C}_8}$.